SOLUTION OF ENERGY EQUATION USING THE INTERVAL BOUNDARY ELEMENT METHOD
Alicja Piasecka Belkhaayt
Department for Strength of Materials and Computational Mechanics Silesian University of Technology, Poland, e-mail: alicja.piasecka@polsl.pl
Abstract. In the paper the 1D energy equation with an interval source function is consi- dered. For this type of equation the 1st scheme of the interval boundary element method is presented. As an example the pure metal crystallization process is analyzed.In the final part of the paper the results of numerical computations for the cooper crystallization process with the interval source function are shown.
1. Interval boundary element method
Transient temperature field in 1D domain describes the following energy equa- tion
( , )
2(0, ): T x t ( , ) ( , )
x L c T x t Q x t
t
∈ ∂ = λ∇ +
∂
% (1)
where c is the volumetric specific heat, λ is the thermal conductivity, Q x t % ( , ) is the interval source function [1], T, x, t denote temperature, spatial co-ordinate and time, respectively.
The above equation must be supplemented by the following boundary-initial conditions
1
2
0
( , )
0: ( , ), 0
( , )
: ( , ), 0
0: ( , ) ( )
T x t
x F T x t
x T x t
x L F T x t
x
t T x t T x
= ∂ =
∂
∂
= =
∂
= =
(2)
The 1
stscheme of the boundary element method has been applied to solve
the problem analyzed [2-5].
At first the time grid is introduced
0 1 2 1
0 = < < < < t t t K t
f−< t
f< < K t
F< ∞ (3) with a certain constant time step
∆ =t tf −tf−1.
The boundary interval integral equation corresponding to the transition
1
f f
t
−→ t is of the form
1
1
1
0
1 1
0 0
0
( , ) 1 ( , , , ) ( , ) d
1 ( , , , ) ( , ) d ( , , , ) ( , ) d
1 ( , ) ( , , , ) d d
f
f
f
f
f
f
t x L
f f
t x
t x L L
f f f f
t x
t L
f
t
T t T x t t q x t t
c
q x t t T x t t T x t t T x t x
c
Q x t T x t t x t c
−
−
−
=
∗
=
=
∗ ∗ − −
=
∗
ξ + ξ =
ξ + ξ +
ξ
∫
∫ ∫
∫ ∫
% %
% %
%
(4)
where ξ is the point where the concentrated heat source is applied, T
∗( , , ξ x t
f, ) t is the fundamental solution, q
∗( , , ξ x t
f, ) t is the heat flux corresponding to the fundamental solution, ( , ) q x t % = −λ∂ T x t % ( , ) / ∂ x is the interval boundary heat flux, ( , ) T x t % is the interval temperature value.
In the case of using the constant elements with respect to time the equation (4) can be written as
1
1
1
0
1 1
0 0
1
0
( , ) 1 ( , ) ( , , , ) d
1 ( , ) ( , , , ) d ( , , , ) ( , ) d
1 ( , ) ( , , , ) d d
f
f
f
f
f
f
t x L
f f f
t x
t x L L
f f f f f
t x
L t
f f
t
T t q x t T x t t t
c
T x t q x t t t T x t t T x t x
c
Q x t T x t t t x
c
−
−
−
=
∗
=
=
∗ ∗ − −
=
− ∗
ξ + ξ =
ξ + ξ +
ξ
∫
∫ ∫
∫ ∫
% %
% %
%
(5)
The numerical approximation of this equation leads to the following interval equation
0 0
1 1
( , ) ( , ) ( , ) ( , ) ( , )
( , ) ( , )
x L
f f x L f
x x
f f
T t g x q x t h x T x t
P t Z t
= =
= =
− −
ξ + ξ = ξ +
ξ + ξ
% % %
% %
(6)
where
1
2
( , ) 1 ( , , , ) d
( )
exp erfc
2 2
4 ( )
f
f
t
f
t
f
g x T x t t t
c
x x
t x
c a t t a t
−
ξ =
∗ξ =
− ξ − ξ
∆ − − ξ − λ
λ − ∆
∫
(7)
and
1
1 sgn( )
( , ) ( , , , ) d erfc
2 2
f
f
t
f
t
x x
h x q x t t t
c
−a t
∗
− ξ − ξ
ξ = ∫ ξ = ∆ (8)
while a = λ / c is the diffusion coefficient.
The interval values P % ( , ξ t
f−1) are defined as
1 1 1
0
2
1
0
( , ) ( , , , ) ( , ) d
1 ( )
exp ( , ) d
2 4
L
f f f f
L
f
P t T x t t T x t x
x T x t x
a t a t
− ∗ − −
−
ξ = ξ =
− − ξ
π ∆ ∆
∫
∫
% %
%
(9)
and the interval values connected with the interval source function take the form
1
1 1
0
1
0
( , ) 1 ( , ) ( , , , ) d d
( , ) ( , ) d
f
f
L t
f f f
t L
f
Z x t Q x t T x t t t x
c
Q x t g x x
−
− − ∗
−
= ξ =
ξ
∫ ∫
∫
%
%
%
(10)
Taking into account the boundary conditions (2) the following system of interval equations is obtained
( )
( )
11 12 11 12
21 22 21 22
1 1
1 1
0, (0, )
( , ) ,
(0, ) (0, )
( , ) ( , )
f f
f f
f f
f f
q t
G G H H T t
G G q L t H H T L t
P t Z t
P L t Z L t
− −
− −
= +
+
% %
% %
% %
% %
(11)
where
11 12
21 2 2
(0, 0) (0, )
( , 0) ( , )
G g G g L
G g L G g L L
= − =
= − = − (12)
and
11 12
21 2 2
0.5 (0, )
( , 0) 0.5
H H h L
H h L H
= − =
= − = − (13)
After determining the ’missing’ boundary values the interval temperatures ( ,
f)
T % ξ t at internal nodes of the domain considered are calculated using the formula
1 1
( , ) ( , ) ( , ) ( , 0) (0, )
( , ) ( , ) ( , 0) (0, ) ( , ) ( , )
f f f
f f f f
T t h L T L t h T t
g L q L t g q t P t
−Z t
−ξ = ξ − ξ −
ξ + ξ + ξ + ξ
% % %
% %
% % (14)
2. Interval source function
The solidification process in one-dimensional domain of pure metal is presented as an example of the interval source function appearing in the mathematical description. It is assumed that the nucleation coefficient and nuclei growth one are interval values and the both coefficients are proportional to the second power of undercooling. The driving force of crystallization is the local and temporary undercooling below solidification point T
cr. The nucleation and nuclei growth are proportional to the second power of undercooling [6, 7].
The interval source function can be defined using the following formula ( , )
( , )
crS x t Q x t Q
t
= ∂
∂
%
% (15)
where Q is the volumetric latent heat, ( , )
crS x t % is the interval volumetric fraction of the solid state at the neighborhood of the point considered x.
In this paper the exponential solidification model proposed by Mehl-Johnson- Avrami-Kolmogoroff is applied
[ ]
( , ) 1 exp ω ( , )
S x t = − − x t (16)
or
4
3( , ) 1 exp π ( , ) ( , ) S x t 3 N x t R x t
= − −
% % % (17)
where N x t % ( , ) is the interval grain density [grains/m
3], R x t % ( , ) is the interval
value of the temporary radius of single grain.
The interval calculations of the source function connected with the crystalliza- tion process modelling require to take into account the interval values of the nucle- ation coefficient γ % = γ, γ and the growth coefficient µ % = µ, µ .
The grain density N x t % ( , ) and the solidification rate ( , ) u x t % are interval values and are defined as follows
( , ) γ ( , )
2N x t % = ∆ % T x t (18)
and
( , ) µ ( , )
2u x t % = ∆ % T x t (19)
where the undercooling ∆ T x t ( , ) is expressed as
1 1
( , ) ( , ) , ( , ) ( , )
2 2
( , )
0, 1 ( , ) ( , )
2
cr cr
cr
T T x t T x t T x t T x t T
T x t
T x t T x t T
− + + ≤
∆ =
+ >
(20)
while T x t and ( , ) T x t denote the first and the second endpoints of the ( , ) temperature interval respectively.
The interval source function is calculated according to the rules of the interval arithmetic [6] and can be expressed as follows
3 2
3
4 ( , )
( , ) π ( , ) 3 ( , ) ( , ) ( , )
3
exp 4 π ( , ) ( , ) 3
cr
N x t
Q x t Q R x t N x t R x t u x t
t
N x t R x t
∂
= ∂ +
× −
% % % % % %
% %
(21)
3. Results of computations
Let us consider the crystallization process proceeding in a copper plate of di- mension L = 0.01 m (one-dimensional problem). The following input data have been introduced: initial temperature T
0= 1120°C, solidification point T
cr= 1083°C, thermal conductivity λ = 280 W/m⋅K, specific heat c = 490 J/kg⋅K, density ρ = 8600 kg/m
3, nuclei coefficient γ % = 10
9− 1000, 10
9+ 1000 1/K
2⋅m
3, growth coefficient
6 6
µ % = 2.95 10 , 3.05 10 ⋅
−⋅
−m/s⋅K
2, volumetric latent heat Q
cr= 1754.4 MJ/m
3. On
the left side of the domain considered the boundary condition of the second type is
assumed: q
b= 0 W/m
2, on the right side the boundary condition of the first type is
assumed: T
b= 1070°C, the domain considered has been divided into 20 constant elements, time step ∆ t = 0.002 s.
Figures 1 and 2 illustrate the cooling curves obtained at the nodes 10 (x = 0.00475 m) and 12 (x = 0.00575 m) of the domain considered, where Tem L, Tem R denote the lower and the upper bounds of the temperature intervals.
1060 1080 1100 1120
0 5 10 15 t[s] 20
T[oC]
Tem L Tem R node 10
Fig. 1. Cooling curves at node 10
1060 1080 1100 1120
0 5 10 15 t[s] 20
T[oC]
Tem L Tem R node 12
Fig. 2. Cooling curves at node 12
Figure 3 presents the courses of the source function at the same nodes, where
Source L and Source R denote the first and the second endpoints of the source
interval. Figure 4 illustrates the temporary interval mean radiuses at the nodes 10 and 12.
0 100 200 300 400 500
0 5 10 15 t[s] 20
[MW/m3]
Source L Source R
node 10 node 12
Fig. 3. The courses of the source function
0 25 50 75 100 125
0,0 1,0 2,0 3,0 t[s] 4,0
R[m]·10-6
Rad L Rad R
node 12
node 10
Fig. 4. The courses of the radius
Summing up, the interval boundary element method is an effective tool in numerical modelling of the problems with the interval source function.
This paper is a part of the project ”Progress and application of identification
methods in moving boundary problems” (No. N507 3592 33).
References
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[2] Brebbia C.A., Telles J.C.F., Wrobel L.C., Boundary Element Techniques, Springer-Verlag 1984.
[3] Majchrzak E., Boundary element method in heat transfer, Publ. of Czestochowa Univ. of Techno- logy, Czestochowa 2001 (in Polish).
[4] Mochnacki B., Suchy J.S., Numerical methods in computations of foundry processes, PFTA, Cracow,1995.
[5]
Piasecka Belkhayat A., Modelling of two-dimensional transient diffusion problem with interval thermal parameters and interval boundary conditions, 17th International Conference on Computer Methods in Mechanics CMM-2007, Short Papers, Łódź-Spała 2007, 311-312.
[6] Kapturkiewicz W., Modelling of cast iron crystallization, Akapit, Cracow 2003 (in Polish).
[7] Majchrzak E., Piasecka A., The numerical micro/macro model of solidification process, Journal of Materials Processing Technology 1997, 64, 267-276.
